Nonconvex gradient flow in the Wasserstein metric and applications to constrained nonlocal interactions

TL;DR

This paper extends the theory of Wasserstein gradient flows to more general energies, providing new well-posedness results, convergence estimates, and applications to constrained nonlocal interactions relevant in biological models.

Contribution

It introduces a broader well-posedness framework for Wasserstein gradient flows based on an Osgood-type criterion and offers the first quantitative convergence estimates outside semiconvex energies.

Findings

Extended well-posedness theory for non-semiconvex energies.

Provided quantitative convergence estimates for JKO scheme.

Applied results to constrained nonlocal interaction energies in biological contexts.

Abstract

Over the past fifteen years, the theory of Wasserstein gradient flows of convex (or, more generally, semiconvex) energies has led to advances in several areas of partial differential equations and analysis. In this work, we extend the well-posedness theory for Wasserstein gradient flows to energies satisfying a more general criterion for uniqueness, motivated by the Osgood criterion for ordinary differential equations. We also prove the first quantitative estimates on convergence of the discrete gradient flow or "JKO scheme" outside of the semiconvex case. We conclude by applying these results to study the well-posedness of constrained nonlocal interaction energies, which have arisen in recent work on biological chemotaxis and congested aggregation.